4 UNIFORMIZABLE SPACES 13 We begin by showing that the topology of E is Hausdorff. Let *, y be two distinct points of E. If there exists an index n such that x e Un and y $ ~Cn , then V = Uw and W = E - Ow are open neighborhoods of x and *y respectively which do not intersect. If on the other hand x e UM and y e Un for some n, then in the metrizable subspace Ort there exists an open neighborhood V0 of x and an open neighborhood W0 of y which do not intersect. The set V = V0 n Un is an open neighborhood of x in E, and there exists an open set W in E such that W n 0M = W0 . Hence W is an open neighborhood of y in E, and V n W = 0. For each n, let dn be a distance defining the topology of Ort , and let (VaJwgji be a basis for the topology of Un, where Vm/J is an open ball with center amn and radius rmn (3.9.4). Letfmn be the function which is equal to rmn - dn(amny x) in VWII and is 0 in the complement of Vmrt in E. Then/mn is continuous on E, since it vanishes at the frontier points of VW/J . Now let for all x, y in E. By virtue of (12.4.6), the proof will be complete if we show that the pseudo-distances dnm define the topology of E. For each x0 e E, every set defined by an inequality of the form dmn(x0 ,x)< a is open in E (3.11.4). Conversely, there exists an integer n such that x0 E Un , and for each neighborhood W of *0 in E there exists an integer W such that Vmn is an open neighborhood of x0 contained in W (3.9.3). Hence fmn(x0) = ft > 0, and the set of all x e E such that dmn(x0 , x) < %fl is therefore contained in W. By virtue of (12.2.1), this completes the proof. We remark that the conclusion of (12.4.7) is not valid without the hypothe- sis of denumerability on the open covering (UA) (Section 12.16, Problem 22); again, the hypotheses cannot be weakened by assuming merely that the open sets Urt are separable and metrizable (section 12.3, Problem 6(f)). PROBLEMS 1. Let E be a topological space such that, for each x e E, the neighborhoods of x which are both open and closed in E form a fundamental system of neighborhoods of x. Show that E is uniformizable. (Observe that the characteristic function of a set which is both open and closed in E is continuous on E.) 2. Let E be a denumerable Hausdorff uniformizable space. Show that, for each x e E, the open-and-closed neighborhoods of x form* a fundamental system of neighborhoods of*. HUNT LIBRARY > ?rrr$«TJB6H, PEMKSYLVAKIA 15211he sets of 93, and argue by con-